Sununu and Shaheen Cruise in New Hampshire Primaries for Governor and Senate

In July, The Washington Post reported that a foundation Mr. Messner started to provide scholarships to underprivileged high school students had awarded just one scholarship in 10 years.

In a Granite State Poll last week, Ms. Shaheen held nearly a 20-point lead over Mr. Messner.

In the Democratic primary for governor, Dan Feltes, the State Senate majority leader, was in a tight race with Andru Volinsky, a lawyer and education activist who was endorsed by Mr. Sanders, the Vermont senator. The support of Mr. Sanders, who won New Hampshire’s presidential primary in February, helped rally progressive voters in a contest that was little noticed compared with other face-offs this year between the Democratic left wing and the party establishment.

Mr. Volinsky broke with New Hampshire Democratic tradition and refused to take “the pledge,” a promise not to introduce sales or state income taxes. “We are the last Bernie candidate standing,” said Irene Lin, Mr. Volinsky’s campaign manager.

Despite being outspent by Mr. Feltes, Mr. Volinsky had a slight two-point lead in the Granite State Poll, within the margin of error.

But again, the primary winner will face a popular incumbent, Mr. Sununu, who has the approval of seven in 10 New Hampshire voters.

In another closely watched race, Matt Mowers, who worked in the Trump administration and received the president’s endorsement, won the Republican primary for the First Congressional District. He will take on Representative Chris Pappas, a Democrat, who is favored to hold the seat by several nonpartisan ratings analysts.

In Rhode Island, which also held primaries on Tuesday, only one of its two congressional districts featured competitive races: Representative Jim Langevin, a 10-term Democrat, eased past a challenger, while on the Republican side Robert Lancia advanced to face Mr. Langevin in November.